1. Field of the Invention
The present invention pertains to means and methods for removing or eliminating the Gibbs phenomenon from Fourier transforms, and more particularly, it pertains to digital techniques or digital processors, such as digital filters or other spectral resolution devices, which filter a train of digital signals or measure the spectral ratios of two trains of digital signals using the Fourier transform method.
2. Description of the Prior Art
In the design of digital filters or other digital devices requiring spectral resolution (e.g., spectral ratio measuring devices to determine the attenuation coefficient or quality factor from signals), a commonly used technique is the discrete Fourier transform (DFT) or the fast Fourier transform (FFT) wherein a train of digital signals in the time domain is transformed into the frequency domain for a selective elimination (filtering) or ratio measurement (e.g., attenuation coefficient or quality factor Q measuring). A continuing problem with such methods is the well-known Gibbs phenomenon, which appears in a DFT (or FFT) due to the incomplete periodicity of the digital signal train within its length which fails to match with the complete periodicity of the sinusoids in the DFT (or FFT) at the end of the signal train. That is to say, the start and end values in the digital signal train provide discontinuities or a difference between its end points which introduce or leak spurious frequency components into the Fourier transformed signals. The conventional way to treat the Gibbs phenomenon is to apply a window function to the signal train to taper it to zero at the end points. However, the use of windowing will distort the waveform of the signals to be measured or induce end effects, i.e., the introduction of spurious frequency components into the signals.